Transport of localized and extended excitations in one-dimensional electrical lattices

TL;DR

This paper analyzes how localized and extended electrical excitations propagate in one-dimensional bi-inductive lattices, deriving closed-form expressions for mean square displacement and transmission coefficients, including Fano resonances, in various impurity configurations.

Contribution

It provides the first analytical derivations of excitation transport and Fano resonance positions in bi-inductive electrical lattices with impurities.

Findings

Ballistic propagation of localized excitations in infinite and semi-infinite lattices.

Closed-form transmission coefficients for plane waves across impurity regions.

Explicit formulas for Fano resonance positions based on coupling strengths.

Abstract

We study the scattering properties of a bi-inductive electrical lattice consisting of a one-dimensional array of coupled LC units. For an initially localized electrical excitation, and in the absence of any impurity, we compute in closed form the mean square displacement of an initially localized electrical excitation for the cases of an infinite and semi-infinite lattice, obtaining a ballistic propagation under very general conditions. For the transport of extended excitations, we compute in closed form the transmission coefficient of electro-inductive plane waves across an impurity region, containing a number of side-coupled units, or a single internal impurity with coupling to first-and second nearest neighbors, looking for the presence of Fano resonances (FRs). For all cases examined, we obtain a closed-form expression for the position of the FR in terms of the relative strengths of the inductive couplings involved. For the case of two, identical side-coupled impurities, the position of the single FR turns out to be independent of the relative distance between the two impurities.